J. Fluid Mech. (2017), vol. 822, pp. 5–30. c© Cambridge University Press 2017This is an Open Access article, distributed under the terms of the Creative Commons Attributionlicence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, andreproduction in any medium, provided the original work is properly cited.doi:10.1017/jfm.2017.212

5

Effect of wall cooling on boundary-layer-inducedpressure fluctuations at Mach 6

Chao Zhang1, Lian Duan1,† and Meelan M. Choudhari2

1Missouri University of Science and Technology, Rolla, MO 65401, USA2NASA Langley Research Center, Hampton, VA 23681, USA

(Received 23 November 2016; revised 21 February 2017; accepted 23 March 2017;first published online 31 May 2017)

Direct numerical simulations of turbulent boundary layers with a nominal free-streamMach number of 6 and a Reynolds number of Reτ ≈ 450 are conducted at awall-to-recovery temperature ratio of Tw/Tr = 0.25 and compared with a previousdatabase for Tw/Tr = 0.76 in order to investigate pressure fluctuations and theirdependence on wall temperature. The wall-temperature dependence of widely usedvelocity and temperature scaling laws for high-speed turbulent boundary layers isconsistent with previous studies. The near-wall pressure-fluctuation intensities aredramatically modified by wall-temperature conditions. At different wall temperatures,the variation of pressure-fluctuation intensities as a function of wall-normal distanceis dramatically modified in the near-wall region but remains almost intact awayfrom the wall. Wall cooling also has a strong effect on the frequency spectrum ofwall-pressure fluctuations, resulting in a higher dominant frequency and a sharperspectrum peak with a faster roll-off at both the high- and low-frequency ends. Theeffect of wall cooling on the free-stream noise spectrum can be largely accountedfor by the associated changes in boundary-layer velocity and length scales. Thepressure structures within the boundary layer and in the free stream evolve lessrapidly as the wall temperature decreases, resulting in an increase in the decorrelationlength of coherent pressure structures for the colder-wall case. The pressure structurespropagate with similar speeds for both wall temperatures. Due to wall cooling, thegenerated pressure disturbances undergo less refraction before they are radiated tothe free stream, resulting in a slightly steeper radiation wave front in the free stream.Acoustic sources are largely concentrated in the near-wall region; wall cooling mostsignificantly influences the nonlinear (slow) component of the acoustic source termby enhancing dilatational fluctuations in the viscous sublayer while damping vorticalfluctuations in the buffer and log layers.

Key words: high-speed flow, turbulence simulation, turbulent boundary layers

† Email address for correspondence: duanl@mst.edu

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6 C. Zhang, L. Duan and M. M. Choudhari

1. IntroductionAn understanding of the physics of pressure fluctuations induced by high-speed

turbulent boundary layers is important to the structural design of hypersonic vehiclesand to the testing and evaluation of hypersonic vehicles in noisy hypersonic facilities.The fluctuating surface pressure on vehicle surfaces is responsible for vibrational loadand may lead to damaging effects such as flutter. The free-stream pressure fluctuationsradiated from the turbulent boundary layer on the nozzle wall of conventionalhypersonic wind tunnels give rise to tunnel noise that has first-order impact onlaminar–turbulent transition in the tunnel. Given that the surface temperaturesof hypersonic flight vehicles are typically significantly lower than the adiabaticwall temperature and that practical hypersonic facilities for testing and evaluatinghypersonic vehicles are designed to have a non-adiabatic turbulent boundary layer onthe nozzle wall, it is of practical importance to investigate wall-temperature effectson hypersonic turbulent boundary layers and their induced pressure fluctuations.

To date, there is limited literature on the effects of wall cooling on high-speedturbulent boundary layers. Most of the available measurements are restricted to basicturbulence quantities, such as the skin friction and Stanton number, and the mean androot mean square (r.m.s.) fluctuations of velocity and temperature (Fernholz & Finley1980; Smits & Dussauge 2006). Existing numerical studies are largely focused on theeffect of wall cooling on the distribution and scaling of velocity fluctuations and therelationships between temperature and velocity fields at a Mach number of 5 or less(Maeder 2000; Duan, Beekman & Martín 2010; Shahab et al. 2011; Chu, Zhang &Lu 2013; Zhang et al. 2014; Hadjadj et al. 2015; Shadloo, Hadjadj & Hussain 2015;Trettel & Larsson 2016). For example, Duan et al. (2010) performed direct numericalsimulations (DNS) of turbulent boundary layers at Mach 5 over a broad range ofwall-to-recovery temperature ratios (Tw/Tr = 0.18–1.0) and focused on assessingthe validity of Morkonvin’s hypothesis in the high-Mach-number cold-wall regime.Zhang et al. (2014) studied the coupling between the thermal and velocity fields ofcompressible wall-bounded turbulent flows and introduced a generalized Reynoldsanalogy that explicitly accounts for finite wall heat flux for cold-wall boundary layers.Hadjadj et al. (2015) and Shadloo et al. (2015) conducted detailed analyses of theeffect of wall temperature on the statistical behaviour of turbulent boundary layersat Mach 2. Bowersox (2009) and Poggie (2015) studied the modelling of turbulentenergy flux in adiabatic and cold-wall turbulent boundary layers. Trettel & Larsson(2016) introduced a new mean-velocity scaling for compressible wall turbulence withheat transfer; this new scaling achieved excellent collapse of the mean-velocity profileat different Reynolds numbers, Mach numbers and rates of wall heat transfer.

As far as the boundary-layer-induced pressure fluctuations are concerned, thebody of available data is even more scarce. Experimental measurements consistlargely of those at the wall using surface-mounted pressure transducers (Kistler &Chen 1963; Fernholz et al. 1989; Beresh et al. 2011). Previous DNS studies ofpressure fluctuations induced by high-speed turbulent boundary layers have focusedon boundary layers with adiabatic or nearly adiabatic walls (Bernardini & Pirozzoli2011; Di Marco et al. 2013; Duan, Choudhari & Wu 2014; Duan, Choudhari &Zhang 2016). To the best of the knowledge of the authors, no existing studies havebeen conducted in the high-Mach-number cold-wall regime that provide the off-wallfluctuating pressure field including the free-stream acoustic pressure fluctuations. Asa result, a comprehensive understanding of the free-stream disturbance field andits dependence on boundary-layer parameters (e.g. free-stream Mach number, walltemperature and Reynolds number) is still lacking.

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 7

M∞ U∞ (m s−1) ρ∞ (kg m−3) T∞ (K)

5.86 869.1 0.0443 55.0

TABLE 1. Free-stream conditions for Mach 6 DNS of turbulent boundary layers. Theworking fluid is assumed to be a perfect gas.

The objective of the current paper is to investigate the dependence of boundary-layer-induced pressure fluctuations on wall temperature for hypersonic Mach numbers.In a previous paper by the present authors (Duan et al. 2016), the successfulapplication of DNS in capturing the global fluctuating pressure field has beendemonstrated for a spatially developing flat-plate nominally Mach 6 turbulentboundary layer with a wall-to-recovery temperature ratio of Tw/Tr= 0.76. A new DNSdataset at Mach 6 with a different wall temperature (Tw/Tr = 0.25) from the previousMach 6 data (Duan et al. 2016) is introduced for the study of wall-temperatureeffects. The effect of wall temperature on single- and multi-point statistics of thecomputed pressure fluctuations at multiple wall-normal locations (including the innerlayer, the log layer, the outer layer and the free stream) is reported, including theintensity, frequency spectra, space–time correlations and propagation velocities.

The remainder of this paper is structured as follows. The flow conditions selectedfor numerical simulation and the numerical method used are outlined in § 2. Section 4is focused on an analysis of statistical and structural features of pressure fluctuations atmultiple wall-normal locations, highlighting their dependence on the wall temperature.The various statistics examined include pressure-fluctuation intensities, power spectraldensities, two-point pressure correlations, propagation speeds and acoustic sources.Conclusions from the study are presented in § 5.

2. Simulation detailsDirect numerical simulations are performed for zero-pressure-gradient cold-wall

turbulent boundary layers with a free-stream Mach number of 5.86. Two DNS cases(M6Tw025 and M6Tw076) with the same free-stream conditions but different walltemperatures are examined, with the M6Tw076 case corresponding to the previoussimulation by Duan et al. (2016). Table 1 outlines the free-stream conditions forthe simulations, including the free-stream velocity U∞, density ρ∞ and temperatureT∞. The free-stream conditions are representative of those at the nozzle exit of thePurdue Mach 6 Quiet Tunnel (BAM6QT) under noisy operation (Schneider 2001;Steen 2010). Table 2 lists the values of the mean boundary-layer parameters at theselected downstream location (xa) for statistical analysis, including the momentumthickness θ , shape factor H= δ∗/θ (where δ∗ denotes the local displacement thickness),boundary-layer thickness δ, friction velocity uτ =

√τw/ρw, viscous length zτ =µw/ρwuτ

and different definitions of the Reynolds number, namely Reθ ≡ ρ∞U∞θ/µ∞,Reτ ≡ ρwuτδ/µw and Reδ2≡ ρ∞U∞θ/µw. Throughout this paper, the subscripts ∞ andw will be used to denote quantities in the free stream and at the wall respectively.The viscosity µ is calculated using Sutherland’s law, µ = C1T3/2/(T + C2), withconstants C1 = 1.458 × 10−6 and C2 = 110.4. The wall temperature Tw for the caseM6Tw076 is similar to that at the nozzle wall of BAM6QT, corresponding to awall-temperature ratio of Tw/Tr ≈ 0.76, with the recovery temperature estimated asTr = T∞(1+ r(γ − 1)M2

∞/2) based on a recovery factor of r = 0.89. Case M6Tw025

has the same free-stream conditions and Reynolds number, Reτ , as case M6Tw076 but

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8 C. Zhang, L. Duan and M. M. Choudhari

Case Tw (K) Tw/Tr Reθ Reτ Reδ2 θ (mm) H δ (mm) zτ (µm) uτ (m s−1) δi (mm)

M6Tw025 97.5 0.25 2121 450 1135 0.199 8.4 3.6 8.0 33.8 1.33M6Tw076 300 0.76 9455 453 1746 0.948 13.6 23.8 52.6 45.1 13.8

TABLE 2. Boundary-layer properties at the station (xa) selected for the analysis of thepressure field (xa = 88.6δi for case M6Tw025 and xa = 54.1δi for case M6Tw076, with δibeing the boundary-layer thickness at the domain inlet).

a lower wall temperature (Tw/Tr ≈ 0.25) which is more likely to be encountered inhigh-altitude flight. Thus, by comparing the results of cases M6Tw025 and M6Tw076,the effect of wall cooling on the pressure fluctuations can be investigated at a fixedReynolds number Reτ .

Wall cooling causes a change in both the boundary-layer thickness and the fluidproperties across the boundary layer. Experiments and numerical data suggest that asingle Reynolds number is not sufficient to characterize the flow (Smits 1991; Lele1994). However, what definition of the Reynolds number is ‘correct’ for assessingthe effects of wall cooling is still an open question, and the choice of that definitionmainly depends on researcher preference and the research objective (Shadloo et al.2015). For instance, out of the few existing DNS studies on the effect of walltemperature, Maeder (2000), Lagha et al. (2011) and Shadloo et al. (2015) havechosen to match Reτ for reporting their data; Duan et al. (2010) and Chu et al.(2013) have chosen to match Reδ2; Shahab et al. (2011) have chosen to matchReθ . In addition, Shadloo et al. (2015) compared the effects of choosing differentdefinitions of the Reynolds number (Reτ , Reδ2, Reθ ) on the turbulence statistics andshowed that Reτ performs best in collapsing the first- and second-order statisticalmoments for boundary layers with different wall heat transfer values. In the currentstudy, we have chosen to match Reτ based partially on the findings of Shadloo et al.(2015). This selection of the Reynolds number is also due to our decisions withregard to grid resolutions and the limited extent of the computational domain.

The details of the DNS methodology, including numerical methods and initialand boundary conditions, have been documented in our previous papers (Duanet al. 2014, 2016). The DNS methodology has been extensively validated againstexperiments and existing numerical simulations for capturing boundary-layer-inducedpressure fluctuations at supersonic/hypersonic speeds (Duan et al. 2014, 2016). Inparticular, the computational predictions for the mean-velocity profiles and surfacepressure spectrum are in good agreement with experimental measurements for caseM6Tw076 (Duan et al. 2016).

Figure 1 shows the computational set-up for case M6Tw025, which parallelsthat of case M6Tw076 documented in Duan et al. (2016). The computationaldomain size and grid resolution are determined based on the lessons learnt fromDuan et al. (2014, 2016), as summarized in table 3. The streamwise length Lx isadjusted to ensure that the turbulence fluctuations are uncorrelated and minimalspurious correlation can be introduced due to the inflow turbulence generation. Thestreamwise domain size is also large enough so that the free-stream acoustic field hasgone through the transient adjustment due to recycled inflow and has re-establishedequilibrium at the downstream location selected for statistical analysis (xa = 88.6δi).It can be shown that the pressure fluctuations both at the wall and in the freestream for case M6Tw025 have become homogeneous in the streamwise direction

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 9

Recyclingplane

Inflowplane

Rescaling

Flow

FIGURE 1. (Colour online) Computational domain and simulation set-up for the DNScase M6Tw025. The reference length δi is the thickness of the boundary layer (basedon 99 % of the free-stream velocity) at the inlet plane. An instantaneous flow is shownin the domain, visualized by the isosurface of the magnitude of the density gradient,|∇ρ|δi/ρ∞ = 0.98, coloured by the streamwise velocity component (with levels from 0to U∞, blue to red). Here, x, y and z are the streamwise, spanwise and wall-normalcoordinates respectively.

Nx ×Ny ×Nz Lx/δi Ly/δi Lz/δi 1x+ 1y+ 1z+min 1z+max

2400× 400× 560 91.7 8.8 57.5 6.42 3.72 0.46 4.75

TABLE 3. Grid resolution and domain size for case M6Tw025. Here, Lx, Ly andLz represent the domain size in the streamwise, spanwise and wall-normal directionsrespectively. The viscous length scale zτ = 8.0 µm corresponds to xa/δi= 88.6. The terms1z+min and 1z+max are the minimum and maximum wall-normal grid spacings for 06 z/δi 68; δi = 1.33 mm.

after x/δi ≈ 60. Uniform grid spacings are used in the streamwise and spanwisedirections. The grids in the wall-normal direction are clustered in the boundarylayer with 1z+ = 0.46 at the wall, and are kept uniform with 1z+ ≈ 5 in the freestream until up to approximately 8δi or 3.3δ, where δi and δ represent the meanboundary-layer thickness based on u/U∞ = 0.99 at the inflow boundary and at theselected downstream location xa respectively. For the selected grid resolution, thewavelength of the highest-frequency spectral components of free-stream pressurefluctuations (corresponding to ωδ∗/U∞ ≈ 15, as shown in § 4.2) is discretized withat least nine points in the streamwise direction and 12 points in the wall-normaldirection.

In the following sections, averages are first calculated over a streamwise window([xa− 0.5δi, xa+ 0.5δi], with xa= 88.6δi, for case M6Tw025 and [xa− 0.9δi, xa+ 0.9δi],with xa=54.1δi, for case M6Tw076) and the spanwise direction for each instantaneousflow field; then, an ensemble average over 312 flow-field snapshots (corresponding

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10 C. Zhang, L. Duan and M. M. Choudhari

to δi/U∞ ≈ 1016 or δ/uτ ≈ 14.6) and over 153 flow-field snapshots (correspondingto δi/U∞ ≈ 240 or δ/uτ ≈ 7.2) is calculated for cases M6Tw025 and M6Tw076respectively. A smaller number of flow-field snapshots was sufficient for caseM6Tw076 because of the larger spanwise domain size (Ly/δi = 15.7) for this casecompared with that for case M6Tw025 (Ly/δi = 8.8). The effect of spanwise domainsize on flow statistics is monitored by comparing case M6Tw076 with an auxiliarysimulation of the same grid resolution but with a narrower span of Ly/δi = 6.26,and negligible difference is observed in the flow statistics of interest. The outflowboundary condition has no influence on boundary-layer profiles within the selectedstreamwise window over which averages are calculated. Statistical convergence forboth cases is verified by calculating averages over varying streamwise window sizesor over a different number of snapshots and by making sure that the differences inflow statistics are negligible (<1 %) among the different data-averaging techniques.Data for free-stream acoustic radiation were not sampled at the same value of z/δfor the two cases. Therefore, comparison of statistical and spectral characteristicswill be made between predictions at z/δ = 2.36 (i.e. z∞ = 2.36δ) for case M6Tw025and z/δ = 2.63 (i.e. z∞ = 2.63δ) for case M6Tw076. Throughout the paper, standard(Reynolds) averages are denoted by an overbar, f , and fluctuations around standardaverages are denoted by a single prime, as f ′= f − f . Negligible differences have beenfound between the standard and density-weighted (Favre) averages for the statisticsreported in this article.

3. Assessment of DNS dataIn this section, the first- and second-moment statistics of the velocity and

temperature fields are reported at the selected downstream location (xa). The data arecompared with published data, especially those of turbulent boundary layers in thehypersonic cold-wall regime.

Figure 2(a) plots the van Driest transformed mean velocity u+VD, which is definedas

u+VD =1uτ

∫ u

0(ρ/ρw)

1/2 du. (3.1)

The mean velocity shows an approximately logarithmic region where u+VD =

(1/k) log(z+) + C upon van Driest transformation. Consistent with the publisheddata by Duan et al. (2010), Shadloo et al. (2015), Modesti & Pirozzoli (2016) andWu et al. (2017), the van Driest-transformed mean velocity shows a shrinking of thelinear viscous sublayer with higher wall cooling, while the additive constant C in thelog law does not seem to be significantly affected. Figure 2(b) shows a significantlybetter collapse of data in both the viscous sublayer and the log layer among thecomputational datasets involving different wall-cooling rates, after the mean velocityand the wall-normal coordinate are transformed according to the proposal by Trettel& Larsson (2016) as

u+TL =

∫ u+

0

(ρ

ρw

)1/2 [1+

12

1ρ

dρdz

z−1µ

dµdz

z]

du+, z∗ =ρ(τw/ρ)

1/2zµ

. (3.2a,b)

Figure 3 plots the streamwise turbulence intensity and the Reynolds shear stress.A significantly improved collapse of data is achieved by Morkovin’s scaling(Morkovin 1962), which takes into account the variation in mean flow properties.

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 11

5

0

10

15

20

25

30

35

100 103102101

5

0

10

15

20

25

30

35

100 103102101

(a) (b)

Shadloo et al. (2015)

M6Tw025M6Tw076Duan et al. (2010) M5T1

Modesti & Pirozzoli (2016)Duan et al. (2010) M5T2

Wu et al. (2017)

M6Tw025M6Tw076Duan et al. (2010) M5T1

Modesti & Pirozzoli (2016)Duan et al. (2010) M5T2

Wu et al. (2017)

FIGURE 2. (Colour online) Mean-velocity profiles transformed according to (a) van Driestand (b) Trettel & Larsson (2016). Symbols:1 (green), Duan et al. (2010) M5T1, M∞= 5,Reτ = 798, Tw/Tr = 0.18;c, Duan et al. (2010) M5T2, M∞ = 5, Reτ = 386, Tw/Tr = 1.0;B, Modesti & Pirozzoli (2016), M∞ = 1.9, Reτ = 448, Tw/Tr = 0.24; 6 (violet red), Wuet al. (2017), M∞ = 4.5, Reτ = 2200, Tw/Tr = 0.22; E, Shadloo et al. (2015), M∞ = 2,Reτ = 507, Tw/Tr = 0.5.

5

–2

–1

0

1

2

3

4

5

–2

–1

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0–2

–1

0

1

2

3

4

–2

–1

0

1

2

3

4(a) (b)

FIGURE 3. (Colour online) Distribution of r.m.s. velocity components as a function ofwall-normal distance. Curves and symbols: —— (red), M6Tw025, M∞ = 5.86, Reτ = 450,Tw/Tr = 0.25; – · – · – (blue), M6Tw076, M∞ = 5.86, Reτ = 453, Tw/Tr = 0.76; – – –, Duanet al. (2010), M∞ = 5, Reτ = 798, Tw/Tr = 0.18; — · · —, Duan et al. (2010), M∞ = 5,Reτ = 386, Tw/Tr = 1.0; 0, Shadloo et al. (2015), M∞ = 2, Reτ = 507, Tw/Tr = 0.5;1 (violet red), Schlatter & Örlü (2010), M∞ ≈ 0, Reτ = 500; u, Peltier, Humble &Bowersox (2016), M∞ = 4.9, Reτ = 1100, Tw/Tr = 0.9.

Morkovin’s scaling brings the magnitudes of the extrema in the compressible casescloser to the incompressible results of Schlatter & Örlü (2010). The better collapse ofdata between cases M6Tw025 and M6Tw076 in figure 3(b) indicates that the effectof wall cooling on fluctuating velocity intensities can be largely accounted for byMorkovin’s scaling. Similarly improved collapse of data is achieved by Morkovin’sscaling for turbulence intensities in the spanwise and wall-normal directions.

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12 C. Zhang, L. Duan and M. M. Choudhari

5

6

7

1

2

3

4

0 0.2 0.4 0.6 0.8 1.0 100 103102101

0.002

0

0.004

0.006DNS

0

1.0

1.5

2.0

0.2 0.4 0.6 0.8 1.0 0

1.0

1.5

2.0

0.2 0.4 0.6 0.8 1.0

(a) (b)

(c) (d )

Mod

ifie

d SR

A

M6Tw025

M6Tw076

0.5 0.5

0 0

FIGURE 4. (Colour online) The coupling between thermal and velocity fields: (a) meantemperature–velocity relation; (b) DNS-predicted turbulent heat flux and the theoreticalmodel of Bowersox (2009); (c) turbulent Prandtl number Prt; (d) modified SRAs of Huang,Coleman & Bradshaw (1995) and Zhang et al. (2014). Curves and symbols: – – –, Duanet al. (2010), M∞ = 5, Reτ = 798, Tw/Tr = 0.18; — · · —, Duan et al. (2010), M∞ = 5,Reτ = 386, Tw/Tr = 1.0;E, Shadloo et al. (2015), M∞ = 2, Reτ = 507, Tw/Tr = 0.5.

As far as the coupling between thermal and velocity fields is concerned, figure 4plots several temperature–velocity scalings for high-speed turbulent boundary layers,including the mean temperature–velocity relation, the turbulent heat flux ρw′T ′, theturbulent Prandtl number Prt ≡ (ρu′w′(∂T/∂z))/(ρw′T ′(∂u/∂z)) and the modifiedstrong Reynolds analogies (SRAs) of Huang et al. (1995) and Zhang et al. (2014).The present spatial DNS results at Mach 6 are generally consistent with thepredictions from several previous studies at lower Mach numbers (Duan et al.2010; Zhang et al. 2014; Shadloo et al. 2015) with regard to the wall-temperaturedependence of the temperature–velocity scalings. In particular, figure 4(a) shows thatstrong wall cooling causes a deviation of the DNS from Walz’s relation (Walz 1969),which is commonly used to relate the mean temperature and velocity as

TT∞=

Tw

T∞+

Tr − Tw

T∞

(u

U∞

)+

T∞ − Tr

T∞

(u

U∞

)2

. (3.3)

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 13

A significantly improved comparison for the cold-wall case (case M6Tw025) isachieved by using the generalized Reynolds analogy of Zhang et al. (2014), in whicha general recovery factor rg is introduced and Tr in equation (3.3) is accordinglyreplaced by Trg as

TT∞=

Tw

T∞+

Trg − Tw

T∞

(u

U∞

)+

T∞ − Trg

T∞

(u

U∞

)2

, (3.4)

where Trg = T∞ + rgU2∞/(2Cp) with rg = 2Cp(Tw − T∞)/U2

∞− 2Prqw/(U∞τw), where

Pr is the molecular Prandtl number and Cp is the heat capacity at constant pressure.Equation (3.4) explicitly accounts for the wall heat flux qw, and it coincides withWalz’s relation in the case of adiabatic walls.

Figure 4(b) shows that the DNS-predicted turbulent heat flux ρw′T ′ compares wellwith the prediction of the theoretical model by Bowersox (2009), consistent with thefinding by Poggie (2015). The DNS-predicted turbulent Prandtl number compares wellwith the computations of Shadloo et al. (2015) and shows singular behaviour near thewall where the correlation w′T ′ is zero (figure 4c). The SRA relates the temperaturefluctuations T ′rms to the streamwise velocity fluctuations u′rms, as given by

T ′rms/T(γ − 1)M2(u′rms/u)

=1

a(1− (∂T t/∂T)), (3.5)

where a= Prt in Huang’s modified SRA (Huang et al. 1995) and a= Prt ≡ Prt(1+wρ ′u′/ρu′w′)/(1+ wρ ′T ′/ρw′T ′) in Zhang’s version of the modified SRA (Zhang et al.2014), and M = u/

√γRT is the local Mach number. Figure 4(d) shows that the

modified SRA of Zhang et al. (2014) gives a slightly improved prediction betweenu′rms and T ′rms than that of Huang et al. (1995).

4. Boundary-layer-induced pressure fluctuations

In this section, the statistical and spectral characteristics of pressure fluctuationsinduced by hypersonic cold-wall turbulent boundary layers are discussed, highlightingtheir dependence on the wall temperature. The pressure statistics analysed includethe fluctuation intensity, frequency power spectral density, space–time correlations andpropagation speed.

The frequency spectrum of the pressure fluctuations is defined as

Φp(ω, x, z)=1

2π

∫∞

−∞

p′(x, y, z, t)p′(x, y, z, t+ τ)e−iωτ dτ , (4.1)

where the overbar indicates an average over the local streamwise window, thespanwise (y) direction and the time (t). Power spectra for case M6Tw025 arecalculated using the Welch method (Welch 1967) with 12 segments and 50 % overlap.A Hanning window is used for weighting the data prior to the fast Fourier transformprocessing. The sampling frequency is approximately 31 U∞/δi (corresponding to20 MHz), and the length of an individual segment is approximately 156 δi/U∞ forcase M6Tw025. The calculation of power spectra for case M6Tw076 follows thatdescribed in Duan et al. (2016). For both cases, the power spectra do not changeupon changing the window function between Hanning and Hamming windows (at

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14 C. Zhang, L. Duan and M. M. Choudhari

least in the reported frequency ranges), and negligible differences are found when thenumber of data segments is varied from eight to 12.

The two-point space–time correlation coefficient of the pressure field is defined as

Cpp(1x, 1y, 1t, x, z, zref )=p′(x, y, zref , t)p′(x+1x, y+1y, z, t+1t)(

p′2(x, y, zref , t))1/2 (

p′2(x+1x, y+1y, z, t+1t))1/2 ,

(4.2)where 1x and 1y are the spatial separations in the streamwise and spanwise directionsrespectively, 1t is the time delay and zref is the wall-normal location at which thecorrelation is computed.

4.1. Root mean square of pressure fluctuationsIn this section, the wall-normal variation of pressure statistics for the cold-wallhypersonic turbulent boundary layer (case M6Tw025) is discussed. The results arecompared with turbulent boundary layers with an adiabatic or nearly adiabatic wallto highlight the effect of wall cooling.

Figures 5(a) and 5(b) show the profiles of the r.m.s. of pressure fluctuationsp′rms normalized by the local wall shear stress τw. For case M6Tw025, p′rms/τwundergoes a rapid increase in magnitude as z→ 0, with p′rms/τw≈ 3.5 at the wall andp′rms/τw ≈ 2.2 at z/δ ≈ 0.08. The magnitude of pressure fluctuation nearly plateausfor 0.08 / z/δ / 0.2. For case M6Tw076 and the DNS results of Bernardini &Pirozzoli (2011), however, a similarly rapid increase in the magnitude of pressurefluctuation with respect to τw as z → 0 is not observed. Instead, the maximumof p′rms/τw is located away from the wall at z/δ ≈ 0.08 (z+ ≈ 25). The peak ofp′rms/τw is approximately 20 % lower in magnitude for case M6Tw076 than forcase M6Tw025. The large difference in p′rms/τw values close to the wall betweenthe turbulent boundary layer with a cold wall (case M6Tw025) and those with anadiabatic or nearly adiabatic wall (case M6Tw076 and that by Bernardini & Pirozzoli(2011)) indicates a strong influence of wall cooling on the pressure fluctuations nearthe wall. The influence of wall cooling on p′rms/τw becomes much weaker in the outerpart of the boundary layer (z/δ > 0.3) and nearly vanishes in the free stream. Outsidethe boundary layer, p′rms/τw approaches a constant value of p′rms/τw ≈ 0.9 for boththe M6Tw025 and M6Tw076 cases. Figures 5(a) and 5(b) show the profiles of r.m.s.pressure fluctuations normalized by the local wall shear stress p′rms/τw. Figures 5(c)and 5(d) further plot the profiles of r.m.s. pressure fluctuations p′rms normalized by thelocal mean (static) pressure p and the free-stream dynamic pressure q∞ = 0.5ρ∞U2

∞

respectively. In contrast to the similar values of p′rms/τw, significantly different valuesof p′rms/p and p′rms/q∞ are shown throughout the boundary layer between casesM6Tw025 and M6Tw076, indicating that the mean shear stress τw is a better scalingfor p′rms than the mean and dynamic pressures that account for the effect of wallcooling.

4.2. Frequency spectra of pressure fluctuationsFigure 6 compares the wall-pressure spectra of cases M6Tw025 and M6Tw076. Thespectra are normalized so that the area under each curve is equal to unity. Forreference, straight lines with slopes of 2, −1, −7/3 and −5 are also included togauge the rate of spectral roll-off across relatively low, mid, mid-to-high overlap and

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 15

0

1

2

3

4

5

0

1

2

3

4

5

0 20 40 60 80 100 120

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 0.5 1.0 1.5 2.0 2.5

0 0.5 1.0 1.5 2.0 2.5 0 0.5 1.0 1.5 2.0 2.5 0

0.2

0.4

0.6

0.8

(a) (b)

(c) (d)

M6Tw025

M6Tw076

Duan et al. (2014)

M6Tw025

M6Tw076

Duan et al. (2014)

M6Tw025

M6Tw076

M6Tw025

M6Tw076

Bernardini & Pirozzoli (2011) Bernardini & Pirozzoli (2011)

FIGURE 5. (Colour online) Pressure-fluctuation r.m.s. profile p′rms as a function of wall-normal distance normalized by (a,b) the local wall shear stress τw, (c) the mean pressurep and (d) the dynamic pressure q∞. Symbols:6, Duan et al. (2014), M∞= 2.5, Reτ = 510,Tw/Tr = 1.0;E, Bernardini & Pirozzoli (2011), M∞ = 4, Reτ = 506, Tw/Tr = 1.0.

high frequencies respectively, according to Bull (1996). The wall-pressure spectrumshows a strong wall-temperature dependence, especially in regions of mid frequencies(i.e. ωδ∗/U∞ > 0.03 and ων/u2

τ < 0.3) and mid-to-high overlap frequencies (i.e.0.3 < ων/u2

τ < 1), and neither the outer scaling (figure 6a) nor the inner scaling(figure 6b) collapses the spectrum between the two DNS cases. Given that thepressure spectrum at mid frequencies is typically attributed to convected turbulencein the logarithmic region and that at mid-to-high overlap frequencies is attributed toeddies in the highest part of the buffer region (20< z+ < 30) (Bull 1996), the largevariation in the wall-pressure spectrum at mid and mid-to-high overlap frequencieswith wall cooling is consistent with the large changes in eddies in buffer and loglayers, as reflected by the differences in r.m.s. pressure values in figure 5. Thedeviation from Kolmogorov’s −7/3 scaling in the overlap region between mid andhigh frequencies is consistent with the findings of Tsuji et al. (2007) and Bernardini,Pirozzoli & Grasso (2011). At both wall temperatures, the wall-pressure spectrumshows a rather weak frequency dependence at the lowest computed frequenciesand exhibits the ω−5 scaling predicted theoretically by Blake (1986) at the highestcomputed frequencies. The premultiplied wall-pressure spectrum for case M6Tw025

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16 C. Zhang, L. Duan and M. M. Choudhari

10–1

10–2

100

10–1

10–2

100

101

10010–1 101 10010–110–2

10010–1 101 10010–110–20

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

M6Tw025M6Tw076

(a) (b)

(c) (d )

FIGURE 6. (Colour online) Comparison of pressure spectra at the wall (z = 0) betweencases M6Tw025 and M6Tw076. The pressure spectrum is normalized so that the areaunder each curve is equal to unity. (a) Log–log plot in outer scale; (b) log–log plot ininner scale; (c) log–linear plot in outer scale; (d) log–linear plot in inner scale. The areaunder each curve is equal to unity. The value of p′rms at the wall is 100.8 Pa for caseM6Tw025 and 44.3 Pa for case M6Tw076.

(figure 6c,d) consists of a sharper peak with a faster roll-off at both high and lowfrequencies compared with case M6Tw076, and wall cooling causes an increase inthe dominant frequency from ωδ∗/U∞ ≈ 4 (ωνw/u2

τ = 0.4 or f δ/U∞ = 1.2) for caseM6Tw076 to ωδ∗/U∞ ≈ 5 (ωνw/u2

τ = 0.6 or f δ/U∞ = 1.7) for case M6Tw025.Regarding the free-stream pressure spectra, figure 7(a) shows that the low-frequency

range of the pressure spectra Φp is relatively insensitive to Tw/Tr when expressed inouter variables, and figure 7(b) shows that the high-frequency portions nearly overlapin inner variables, which conforms to the findings of the wall-pressure spectrum inlow-speed adiabatic flows (Bull 1996). Moreover, figures 7(c) and 7(d) show thatthe peak of the premultiplied spectrum is centred at a frequency of ωδ∗/U∞ ≈ 1.5,which is more than three times lower than that of the pressure spectrum at the wall,indicating that the characteristic frequency of the acoustic mode is significantly lowerthan that of the vortical fluctuation close to the surface. The dominant frequencyof the free-stream pressure spectrum is independent of wall temperature, indicatingrelatively insignificant influence of wall cooling on the free-stream pressure spectrum.

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 17

10010–1 101

10010–1 101

10010–110–2

10010–110–2

0.1

0

0.2

0.3

0.4

0.5

0.1

0

0.2

0.3

0.4

0.5

100

10–1

10–2

10–3

101

100

10–1

10–2

M6Tw025M6Tw076

(a) (b)

(c) (d )

FIGURE 7. (Colour online) Comparison of pressure spectra in the free stream (z = z∞)between cases M6Tw025 and M6Tw076: (a) log–log plot in outer scale; (b) log–log plotin inner scale; (c) log–linear plot in outer scale; (d) log–linear plot in inner scale. Thearea under each curve is equal to unity. The value of p′rms in the free stream is 24.8 Pafor case M6Tw025 and 13.9 Pa for case M6Tw076.

4.3. Spatial correlation of pressure fluctuationsTo illustrate the spatial size and orientation of statistically significant three-dimensional(3D) pressure structures, figure 8 plots the 3D correlation coefficient of the pressuresignal Cpp(1x, 1y, 0, xa, z, zref ) as a function of wall-normal distance. For eachreference height zref , there exists a downward-leaning pressure structure with finitespatial size and an inclined orientation. The pressure structure has a spatial lengthscale of the order of the boundary-layer thickness O(δ) in each direction and increasesin size as the distance from the wall increases. The pressure structure is approximatelyperpendicular to the direction of U∞ at the wall and becomes increasingly moredownward leaning as it moves away from the wall in the inner and outer regionsof the boundary layer. In the free stream, the inclination angle with respect tothe direction of U∞ approaches θxz ≈ 28. The free-stream wave-front orientationclosely matches the wave-front orientation of the instantaneous acoustic radiationvisualized by numerical schlieren image, as will be shown in figure 9. Consistentwith the spatial correlation in the free-stream region (figure 8d), the 3D visualization

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18 C. Zhang, L. Duan and M. M. Choudhari

00.20.40.60.81.0

00.20.40.60.81.0

00.20.40.60.81.0

1.0

1.5

2.0

2.5

3.0

1

–10

1

–1

0 1

–10

00.5

1.0

–1.0–0.5

00.5

1.0

–1.0–0.5

00.5

1.0

–1.0–0.5

00.5

1.01.5

–1.5–1.0

–0.50

0.51.0

–1.5–1.0

–0.5

xy

z

xy

z

xy

z

xy

z

Flow Flow

FlowFlow

(a) (b)

(c) (d)

FIGURE 8. (Colour online) Three-dimensional representation of the spatial correlationcoefficient Cpp(1x,1y,0, xa, z, zref ) of the pressure signal at multiple wall-normal locationsfor case M6Tw025. The flow goes from left to right towards the positive x direction.Three-dimensional isosurfaces are shown at Cpp = 0.1 (blue) and 0.6 (green). In thehorizontal planes going through the correlation origin (z= zref ), the contour lines shownin white range from 0.1 to 0.9.

in figure 9 shows that the free-stream pressure waves deviate from purely planarbehaviour in the spanwise wall-normal (y–z) plane and exhibit a preferred orientationof θ ≈ 28 in the streamwise wall-normal (x–z) plane. The finite spanwise extent ofthe free-stream pressure waves is consistent with the finite size of acoustic sourcesthat are responsible for generating the waves. Similar patterns of free-stream acousticradiation are also found for case M6Tw076 (Duan et al. 2016).

Figure 10 compares the spatial correlation coefficient (with zero spanwise separation,1y= 0) in the streamwise wall-normal plane between cases M6Tw025 and M6Tw076.At the wall (zref /δ = 0), the pressure structures have a similar inclination angle ofθxz≈81 for both cases. In the free stream, the structure angle for cases M6Tw025 andM6Tw076 decreases to θxz≈ 28 and θxz≈ 21 respectively. The change in inclinationmight indicate that pressure disturbances generated within the boundary layer undergoless refraction before they are radiated to the free stream, resulting in a higher waveangle for case M6Tw025. The reduction in refraction for case M6Tw025 may be dueto the less drastic variation in fluid properties (such as fluid density and temperature)because of wall cooling.

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 19

Flow

xy

z

FIGURE 9. (Colour online) Instantaneous flow visualization for case M6Tw025. The greycontours are those of numerical schlieren, with density gradient contour levels selected toemphasize disturbances in the free stream. The colour contours are those of the magnitudeof vorticity, with contour levels selected to emphasize the large-scale motions within theboundary layer. The angle θ is between the flow direction and the acoustic wave front.

0 0.5 1.0 1.5–1.5 –1.0 –0.5 0 0.5 1.0 1.5–1.5 –1.0 –0.5

0.5

0

1.0

1.5

2.0

2.5

1.5

2.0

2.5

3.0

3.5(a) (b)

FIGURE 10. (Colour online) Contours of the spatial correlation coefficient of the pressuresignal Cpp(1x, 0, 0, xa, z, zref ) in the streamwise wall-normal plane: (a) zref = 0 (wall);(b) zref = z∞ (free stream); ——, M6Tw025; — · —, M6Tw076. Four contour levels areshown: Cpp = 0.1, 0.2, 0.4 and 0.8.

4.4. Propagation and evolution of pressure structuresTo quantify the overall propagation speed of pressure-carrying eddies or wavepacketsas a function of distance from the wall, the bulk propagation speed is obtained as

Ub ≡−(∂p/∂t)(∂p/∂x)

(∂p/∂x)2. (4.3)

This expression defines the bulk propagation speed Ub by finding the value of Ub thatminimizes the difference between the real time evolution of p(x, t) and a propagatingfrozen wave p(x−Ubt). A figure of merit for the frozen-wave approximation can be

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20 C. Zhang, L. Duan and M. M. Choudhari

0 0.5 1.0 1.5 2.0 2.5 100 103102101 0.4

0.6

0.8

1.0

0.4

0.6

0.8

1.0

M6Tw025M6Tw076

(a) (b)

FIGURE 11. (Colour online) Comparison of the bulk propagation speed of pressurefluctuations in (a) outer and (b) inner units between cases M6Tw025 and M6Tw076. Here,Ub is defined based on (4.3).

further defined as

γp ≡

∣∣(∂p/∂x)(∂p/∂t)∣∣[

(∂p/∂t)2 (∂p/∂x)2]1/2 , (4.4)

where γp equals unity for a perfectly frozen wave and is zero for fast decaying ordeforming waves as they convect downstream. This definition of the bulk propagationspeed and figure of merit for the frozen-wave approximation was first used by DelÁlamo & Jiménez (2009) for the streamwise velocity fluctuations in turbulent channelflows.

Figure 11 shows a comparison of the bulk propagation speed Ub between casesM6Tw025 and M6Tw076. Wall cooling has a small influence on the propagationspeed of pressure structures within the main part of the boundary layer and hasnearly no influence on the propagation speed of radiated pressure waves in the freestream. Consistent with previous findings (Duan et al. 2014, 2016), the free-streampropagation speed for case M6Tw025 is significantly lower than the mean velocity inthe free stream.

Figure 12 shows the wall-normal distribution of γp which provides a figure ofmerit for the frozen-wave approximation for cases M6Tw025 and M6Tw076. Forboth wall-temperature conditions, γp is close to unity across the boundary layer,indicating that the propagation effect is overall more dominant than the evolutioneffect for the pressure structures. As the wall temperature decreases, the pressurestructures become more ‘frozen’, with less significant evolution as they propagatedownstream, especially for the pressure structures in the free stream.

The propagation and evolution of large-scale pressure structures can be furtherinvestigated via the space–time correlation contours of pressure fluctuations,Cpp(1x, 0, 1t, xa, zref , zref ). Figure 13 shows contours of constant space–timecorrelation Cpp(1x, 0, 1t, xa, zref , zref ) at the wall (zref = 0) and in the free stream(zref = z∞) for cases M6Tw025 and M6Tw076. The skewed shape of the contoursat both locations indicates the propagative nature of the pressure field, which ischaracterized by downstream propagation of either the coherent pressure-carrying

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 21

0 0.5 1.0 1.5 2.0 2.50.85

0.90

0.95

1.00

M6Tw025M6Tw076

FIGURE 12. (Colour online) The distribution of the correlation coefficient γp whichprovides a figure of merit for the frozen-wave approximation. Here, γp is defined basedon (4.4).

–1

0

1

2

3

–2

–3

10 2 3 4–1–2–3–4

–1

0

1

2

3

–2

–3

10 2 3 4–1–2–3–4

(a) (b)

FIGURE 13. (Colour online) Contours of constant space–time correlation coefficient of thepressure signal Cpp(1x, 0, 1t, xa, zref , zref ): (a) at the wall; (b) in the free stream; ——,M6Tw025; — · —, M6Tw076. Four contour levels are shown: Cpp= 0.1, 0.2, 0.4 and 0.8.

eddies within the boundary layer or the pressure wavepackets in the free stream.Based on the space–time correlation data, the speed of propagation of pressurefluctuations can be estimated as the ratio 1x/1t for a given time delay 1t at thevalue of 1x where

∂C(rx, 0, 1t, xa, zref , zref )

∂rx

∣∣∣∣rx=1x

= 0, (4.5)

or for a given streamwise separation 1x at the value of 1t where

∂C(1x, 0, rt, xa, zref , zref )

∂rt

∣∣∣∣rt=1t

= 0. (4.6)

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22 C. Zhang, L. Duan and M. M. Choudhari

0

0.5

0

1.0

1.5

2.0

0.5

0

1.0

1.5

2.0

2 4 6 2 4 6

(a) (b)

FIGURE 14. (Colour online) Bulk propagation speeds of the pressure fluctuation as afunction of free-stream Mach number: (a) at the wall (zref = 0); (b) in the free stream(zref = z∞). Symbols: squares, Kistler & Chen (1963); left triangles, Bernardini & Pirozzoli(2011); diamonds, Laufer (1964); letters A, B, C, Duan et al. (2014); up triangle, circle,down triangle, case M6Tw025; letters D, E, F, case M6Tw076. Here, Ub1, Ub2 and Ub3are defined based on (4.5), (4.6) and (4.3) respectively.

Figures 14(a) and 14(b) compare the variation of bulk propagation speed withthe free-stream Mach number at the wall and in the free stream respectively withsome existing experiments and simulations. In the figure, Ub1 is defined based onthe space–time correlation coefficient, with (4.5) for the time delay 1t or frequency(ω= 2π/1t) where the premultiplied frequency spectrum (figures 6 and 7) attains itsmaximum. In analogy, Ub2 is derived based on (4.6) for the streamwise separation 1xor wavenumber (k1 = 2π/1x) where the premultiplied one-dimensional wavenumberspectrum attains its maximum. The value of Ub3 is computed using (4.3) by assuminga ‘frozen wave/eddy’. Consistent with figure 11, the propagation speed based onthe space–time correlation coefficient is comparable between cases M6Tw025 andM6Tw076, indicating that wall cooling has only a small influence on the overallpropagation speed of pressure structures away from the wall. The Mach-numberdependence of the bulk propagation speed is consistent with the previous datareported by Bernardini & Pirozzoli (2011) for Ub at the wall and by Laufer (1964)and Duan et al. (2014, 2016) for Ub in the free stream.

To study the propagation speed of spectral components of pressure fluctuations, thephase speed of pressure fluctuations is defined as

Up(ω)=ω1x/θp(ω), (4.7)

where 1x is the distance between two pressure signals separated in the streamwisedirection and θp(ω) is the phase difference between the two streamwise-separatedpressure signals derived based on the cross-spectrum of the two signals. In thecurrent study, the streamwise separation 1x is chosen to be the smallest streamwisedistance at which two pressure signals are spatially sampled (1x+= 6.42 and 28.9 forcases M6Tw025 and M6Tw076 respectively). At the selected streamwise separation,the coherence between the two signals is close to unity and the definition (4.7) thusprovides a ‘local’ measurement of the phase speed. This definition was first used by

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 23

0.4

0.6

0.8

1.0

0 5 10 150.4

0.6

0.8

1.0

0 5 10 15

M6Tw025M6Tw076

(a) (b)

FIGURE 15. (Colour online) Comparison of phase speed (a) at the wall and (b) in the freestream. The phase speed Up(ω) is defined based on equation (4.7). The vertical dashedline denotes the peak frequency ωpk where the premultiplied frequency spectrum attainsits maximum.

Stegen & Van Atta (1970) to measure the local phase speed of the Fourier componentsof the longitudinal velocity fluctuations in grid turbulence with a small probe spacing.Figure 15 shows the phase speed of pressure fluctuations Up(ω) at the wall and inthe free stream. At the wall, the phase speed shows a weak frequency dependence forboth cases, and the wall-pressure structures of all frequencies propagate with speedssimilar to the local bulk propagation speed. In the free stream, while the phase speedof the dominant pressure structures is similar to the local bulk propagation speed,wall cooling slightly increases the free-stream phase speed at higher frequencies, andthe high-frequency pressure structures propagate with a speed larger than the bulkpropagation speed.

To interpret the Lagrangian decorrelation length of the coherent pressure structures,Figure 16 compares the spatial decay of the maximum space–time correlation ofpressure fluctuations, (Cpp)max, at the wall and in the free stream for cases M6Tw025and M6Tw076. The slower rate of spatial decay in (Cpp)max for case M6Tw025indicates that wall cooling de-energizes pressure structures, making them evolve lessrapidly as they propagate downstream. Such a trend is consistent with the largervalues of the ‘frozen-wave’ index γp (figure 12) for case M6Tw025.

4.5. Free-stream acoustic radiationIn this section, the nature of free-stream acoustic fluctuations radiated from theturbulent boundary layer is analysed, including the modal compositions and theacoustic sources.

4.5.1. Modal compositions of free-stream fluctuationsThe characteristics of free-stream fluctuations are analysed using the theory of

modal analysis, which was initially proposed by Kovasznay (1953). Accordingto Kovasznay, the fluctuations at any point within a uniform mean flow can berepresented as a superposition of three different modes with covarying physicalproperties: the vorticity mode, the acoustic or sound-wave mode and the entropymode (also referred to as entropy spottiness or temperature spottiness).

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24 C. Zhang, L. Duan and M. M. Choudhari

10 2 3 4 10 2 3 4

1.0

0

0.2

0.4

0.6

0.8

1.0

0

0.2

0.4

0.6

0.8

M6Tw025: wallM6Tw076: wall

M6Tw025: free streamM6Tw076: free stream

(a) (b)

FIGURE 16. (Colour online) Comparison of the maximum space–time correlationcoefficient of pressure fluctuations, (Cpp)max, as a function of streamwise separation 1x(a) at the wall and (b) in the free stream for cases M6Tw025 and M6Tw076.

M6Tw076 M6Tw025

u′rms/u 1.36× 10−3 2.34× 10−3

v′rms/u 1.05× 10−3 1.62× 10−3

w′rms/u 2.05× 10−3 3.20× 10−3

p′rms/p 2.05× 10−2 3.47× 10−2

ρ ′rms/ρ 1.46× 10−2 2.48× 10−2

T ′rms/T 5.89× 10−3 9.89× 10−3

(ρu)′rms/ρu 1.38× 10−2 2.29× 10−2

T ′t,rms/T t 1.98× 10−3 3.08× 10−3

p′t,rms/pt 6.69× 10−3 1.08× 10−2

(∂u′i/∂xi)2/Ω′jΩ′j 31 580 12 153

s′rms/R 2.11× 10−3 2.29× 10−4

u′p′/u′rmsp′

rms −0.653 −0.829v′p′/v′rmsp

′

rms −0.00639 −0.00512w′p′/w′rmsp

′

rms 0.925 0.956ρ ′p′/ρ ′rmsp

′

rms 1 1T ′p′/T ′rmsp

′

rms 1 1

TABLE 4. The free-stream disturbance field for cases M6Tw025 and M6Tw076. Here, Ris the gas constant in the ideal-gas equation of state p= ρRT .

Table 4 lists the free-stream values of several fluctuating flow variables for casesM6Tw025 and M6Tw076. Here, s is the specific entropy, Ω is the vorticity and thesubscript ‘t’ denotes stagnation quantities. A comparison of the data in the two casesindicates that the magnitude of free-stream fluctuations normalized by the respectivemean values increases significantly as the wall temperature decreases, includingboth the velocity fluctuations and the fluctuations in thermodynamic variables. Inparticular, the pressure fluctuations in the free stream, including p′rms/p and p′t,rms/pt,are considerably different for the two cases (3.47 % versus 2.05 % for p′rms/p, 1.08 %

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 25

versus 0.669 % for p′t,rms/pt, larger values for the colder-wall case). However, p′rms/pand p′t,rms/pt bear nearly the same ratio of approximately 1.7 across the two cases. Forboth wall-temperature cases, the variation in r.m.s. amplitudes of velocity fluctuationsalong the three Cartesian axes indicates the anisotropy of the free-stream velocityfluctuations, with the wall-normal component of the velocity fluctuations being thelargest among the three. The relative perturbations in thermodynamic variables arenearly an order of magnitude larger than the velocity fluctuations and nearly satisfyisentropic relations, indicating the acoustic nature of the free-stream fluctuations. Thedominance of the acoustic model is also indicated by the large ratio of the dilatationalfluctuations (∂u′i/∂xi)2 to the vortical fluctuations Ω ′jΩ ′j and the small values of theentropy fluctuations s′rms/R compared with the pressure fluctuations p′rms/p.

Laufer (1964) had assumed the u′ and p′ fluctuations to be perfectly anticorrelatedduring the reduction of his hot-wire measurements based on the assumption of purelyplanar acoustic waves. However, the numerical simulations for both values of surfacetemperature ratio show that the correlation coefficient between u′ and p′ is differentfrom −1. Cooling of the surface leads to a correlation coefficient of −0.829 forcase M6Tw025, which is closer to −1 than the correlation coefficient of −0.653for case M6Tw076. The less significant deviation from purely planar behaviour forcase M6Tw025 may indicate that acoustic radiation becomes closer to planar acousticwaves with increased wall cooling.

4.5.2. Acoustic sourcesTo understand the effect of wall cooling on the pressure field, an analysis

following Phillips (1960) has been carried out to study the acoustic sources thatare responsible for the pressure fluctuations induced by the turbulent boundary layer.The acoustic source terms can be derived by rearranging the Navier–Stokes equationsinto the form of a wave equation, after neglecting the diffusive terms, as

D2

Dt2−

∂

∂xia2 ∂

∂xi

log(

pp0

)= γ S, (4.8)

where S≡ (∂ui/∂xj)(∂uj/∂xi) is the acoustic source term, which is quadratic in the totalflow velocity, p0 is a convenient reference pressure, D/Dt is the substantial derivativebased on mean flow velocity and γ is the specific heat ratio. The terms on the left-hand side of (4.8) are those of a wave equation in a medium moving with the localmean velocity of the flow. The acoustic source term S on the right-hand side canbe further decomposed into its linear (rapid) component 2(∂U/∂z)(∂w′/∂x) and itsnonlinear (slow) component (∂u′i/∂xj)(∂u′j/∂xi). The details about the acoustic analogyequation, the definition and the decomposition of acoustic source terms are discussedin our previous papers (Duan et al. 2014, 2016).

Figure 17 plots the r.m.s. of the acoustic source term, S′rms, and its linearand nonlinear components in the near-wall region of the boundary layer againstwall-normal distance. For both wall temperatures, the near-wall variation of thetotal acoustic source term conforms well with that of p′rms (figure 5b). For caseM6Tw076, the nonlinear source term is dominant over the linear term throughout theboundary layer (figure 17a), and (∂v′/∂z)(∂w′/∂y) has the largest r.m.s. value amongthe constituent terms of the nonlinear acoustic source (figure 17c). The dominanceof (∂v′/∂z)(∂w′/∂y) may be indicative of the important role played by streamwisevortical structures in sound generation (Duan et al. 2016).

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26 C. Zhang, L. Duan and M. M. Choudhari

50 100 150 2000

50 100 150 2000

50 100 150 2000

0.02

0.01

0.03

0.04

0.05

0.06

0.02

0.01

0.03

0.04

0.05

0.06

0.02

0.01

0.03

0.04

0.05

0.06

M6Tw025,

M6Tw076,M6Tw025,

M6Tw076,

M6Tw076, total

M6Tw076 M6Tw025

Constituent terms of NLS

M6Tw076, NLSM6Tw076, LS

M6Tw025, totalM6Tw025, NLSM6Tw025, LS

R.m

.s. o

f so

urce

term

s

R.m

.s. o

f no

nlin

ear

sour

ce te

rms

(a) (b)

(c)

FIGURE 17. (Colour online) Profiles of the r.m.s. source terms (including the total,nonlinear source (NLS) and linear source (LS) terms) across the near-wall portion of theboundary layer. The r.m.s. values of the source terms are normalized by (νw/u2

τ )2.

As the wall temperature is decreased, the r.m.s. of the nonlinear acoustic term issignificantly reduced in the buffer layer due to the damping of (∂v′/∂z)(∂w′/∂y), andthe linear source term becomes relatively more dominant in this region (figure 17b). Inthe meantime, the r.m.s. value of the nonlinear acoustic term is dramatically increasedin the viscous sublayer, with (∂w′/∂z)2 becoming the most dominant term in thisregion (figure 17c). Given that (∂w′/∂z)2 is related to the dilatational fluctuationsof velocity and (∂v′/∂z)(∂w′/∂y) is related to the near-wall streamwise vorticalfluctuations, the variation of these terms with wall temperature may indicate that wallcooling influences sound generation largely by enhancing dilatational motions in theviscous sublayer while damping streamwise vortical structures in the buffer layer. Theenhancement of the dilatational motions in the viscous sublayer and the damping ofthe streamwise vortical structures in the boundary layer are also apparent from therapid increase in r.m.s. dilation and r.m.s. streamwise vorticity near the wall, as seenfrom figures 18(a) and 18(b). The enhancement of dilatational motions near the wallis not unexpected as wall cooling increases the turbulent Mach number by causing adecrease in the local sound speed.

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Effect of wall cooling on boundary-layer-induced pressure fluctuations 27

50 100 150 2000 50 100 150 2000

0.06

0.04

0.02

0

0.08

0.2

0.1

0

0.3

M6Tw025M6Tw076

M6Tw025M6Tw076

(a) (b)

FIGURE 18. (Colour online) Profiles of the r.m.s. of dilatation and streamwise vorticityacross the near-wall portion of the boundary layer normalized using νw/u2

τ .

0 5 10 15

0.2

0.4

0.6

0.8

1.0M6Tw025M6Tw076

FIGURE 19. (Colour online) The phase speed of the acoustic source term. Here, Us(ω)is defined based on (4.7) for the acoustic source term S.

Figure 19 compares the phase speed derived from the acoustic source term, Us(ω),between cases M6Tw025 and M6Tw076 in the buffer layer. Wall cooling increasesthe convection speed of the acoustic sources for all frequencies. At the dominantfrequency of free-stream acoustic radiation, ωpkδ/U∞= 1.5, which corresponds to thepeak frequency of the premultiplied spectrum shown in figure 7(c), the convectionspeed of the acoustic source is 0.64U∞ (Mr = 2.11) and 0.55U∞ (Mr = 2.64)respectively for cases M6Tw025 and M6Tw076, with Mr ≡ (U∞ − Us)/a∞. Thefact that acoustic sources propagate supersonically with respect to the free stream isconsistent with the concept of ‘eddy Mach wave’ radiation (Phillips 1960). Giventhat the radiation wave angle can be approximated via the ‘Mach angle’ relation as1/ sin θ =Mr, the smaller value of Mr for case M6Tw025 is consistent with the largerradiation wave angle of 28 for this case (figure 10b).

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28 C. Zhang, L. Duan and M. M. Choudhari

5. Summary and conclusionsDirect numerical simulations of Mach 5.86 turbulent boundary layers with two

wall temperatures (Tw/Tr = 0.25, 0.76) are compared to investigate the effect of wallcooling on the pressure fluctuations generated by hypersonic turbulent boundary layers.The simulations show that wall cooling significantly modifies the pressure-fluctuationintensities near the wall, with p′w,rms/τw varying from 2.8 for Tw/Tr = 0.76 to 3.5for Tw/Tr = 0.25. Furthermore, the frequency spectra of the wall-pressure fluctuationsfor the two cases show considerable differences when plotted in terms of eitherouter-layer or inner-layer variables. The peak of the premultiplied spectrum shiftsto a higher value as the wall temperature decreases. Wall cooling slows downthe evolution of pressure wavepackets at the wall, resulting in a larger decorrelationlength of pressure structures, but has little influence on the bulk propagation speeds ofwall-pressure structures. Regarding the free-stream pressure fluctuations, although theintensity shows a strong wall-temperature dependence when normalized by the meanfree-stream pressure (p

∞), it compares well between the two cases when normalized

by the local wall shear stress τw. The frequency spectra of free-stream radiationcollapse well between the two cases when normalized in terms of outer or innerboundary-layer parameters. Wall cooling results in an increase in the radiation waveangle (defined based on spatial correlations, Cpp) from 21 for Tw/Tr = 0.76 to 28for Tw/Tr = 0.25. Similarly to pressure structures at the wall, the free-stream pressurestructures evolve less rapidly as the wall temperature decreases. The propagation speedof free-stream pressure structures is found to be insensitive to the wall temperatureand is significantly smaller than the free-stream velocity for both cases. An analysisof acoustic sources using the acoustic analogy of Phillips (1960) shows that wallcooling influences sound generation largely by enhancing dilatational motions in theviscous sublayer while damping streamwise vortical structures in the buffer layer.

AcknowledgementsThis material is based on work supported by the Air Force Office of Scientific

Research through award no. FA9550-14-1-0170, managed by Dr I. Leyva. Thework was initiated with support of the NASA Langley Research Center under theResearch Cooperative agreement no. NNL09AA00A (through the National Instituteof Aerospace). Computational resources were provided by the NASA AdvancedSupercomputing Division, the DoD High Performance Computing ModernizationProgram and the NSF’s PRAC program (NSF ACI-1640865).

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